Strict Ordering on Integers is Well-Defined

Theorem

Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers.

Let:

\(\ds \eqclass {a, b} {}\) \(=\) \(\ds \eqclass {a', b'} {}\)
\(\ds \eqclass {c, d} {}\) \(=\) \(\ds \eqclass {c', d'} {}\)

Then:

\(\ds \eqclass {a, b} {}\) \(<\) \(\ds \eqclass {c, d} {}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \eqclass {a', b'} {}\) \(<\) \(\ds \eqclass {c', d'} {}\)


Proof

This is a direct application of the Extension Theorem for Total Orderings.

$\blacksquare$


Sources

  • 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers
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